Discretization of a New Method for Localizing Discontinuity Lines of a Noisy Two-Variable Function

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Abstract

We consider the ill-posed problem of localizing (finding the position of) discontinuity lines of a noisy function of two variables. New regularizing methods of localization are constructed in a discrete form. In these methods, the smoothing kernel is varying, which simplifies the implementation of the algorithms. We obtain bounds for the localization error of the methods and for their separability threshold, which is another important characteristic.

Keywords

ill-posed problem localization of singularities line of discontinuity regularization discretization. 

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References

  1. 1.
    S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1999; Mir, Moscow, 2005).MATHGoogle Scholar
  2. 2.
    Ya. A. Furman, A. V. Krevetskii, A. K. Peredreev, A. A. Rozhentsov, R. G. Khafizov, I. L. Egoshina, and A. N. Leukhin, Introduction to Contour Analysis and Its Applications to Image and Signal Processing (Fizmatlit, Moscow, 2002) [in Russian].Google Scholar
  3. 3.
    A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1974; Wiley, New York, 1981).MATHGoogle Scholar
  4. 4.
    V. V. Vasin and A. L. Ageev, Ill-Posed Problems with A Priori Information (VSP, Utrecht, 1995).CrossRefMATHGoogle Scholar
  5. 5.
    A. L. Ageev and T. V. Antonova, “On a new class of ill-posed problems,” Izv. Ural. Gos. Univ., No. 58, 24–42 (2008).MathSciNetMATHGoogle Scholar
  6. 6.
    A. L. Ageev and T. V. Antonova, “On ill-posed problems of localization of singularities,” Trudy Inst. Mat. Mekh. UrO RAN 17 (3), 30–45 (2011).Google Scholar
  7. 7.
    A. L. Ageev and T. V. Antonova, “New methods for the localization of discontinuities of the first kind for functions of bounded variation,” J. Inverse Ill-Posed Probl. 21 (2), 177–191 (2013).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    T. V. Antonova, “Localization method for lines of discontinuity of an approximately defined function of two variables,” Numer. Anal. Appl. 5 (4), 285–296 (2012).CrossRefMATHGoogle Scholar
  9. 9.
    A. L. Ageev and T. V. Antonova, “Approximation of discontinuity lines of a noisy function of two variables,” J. Appl. Ind. Math. 6 (3), 269–279 (2012).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    A. L. Ageev and T. V. Antonova, “On discretization of methods for localization of singularities of a noisy function,” Trudy Inst. Mat. Mekh. UrO RAN 21 (1), 3–13 (2015).MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

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